The 22-year-old Pole, who won the French Open title for the third time last month to make it four grand slam crowns, has yet to get past the fourth round of the London major.
Zhu, ranked 34th in the world, earned a break point in the first game but Swiatek won 11 points in a row to sprint into a 3-0 lead.
Another break put Swiatek, a renowned slider on clay and hard courts who has often struggled with her movement on grass, firmly in the driving seat.
Zhu saved two set points at 5-0 but Swiatek sealed the first set a game later with a powerful crosscourt forehand winner.
The pair traded breaks early in the second before Swiatek, who pulled out of her Bad Homburg semi-final on Friday due to illness, earned a second one and was 4-3 up when play was interrupted due to rain.
It resumed after the roof on Court One was shut but Swiatek needed only seven minutes to win the two games she needed and sealed victory with a backhand winner.Â
Twice major winner Victoria Azarenka held off China's Yuan Yue in a tight match to secure a 6-4 5-7 6-4 win and advance to the second round.
American fourth seed Jessica Pegula overcame a second-set wobble to beat compatriot Lauren Davis 6-2 6-7(8) 6-3 on Monday.
France's Diane Parry dealt Briton Harriet Dart a second set bagel en route to a 6-7(4) 6-0 6-4 comeback win.
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